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Let $\Gamma(\mathcal{H})$ be the boson Fock space over a finite dimensionalHilbert space $\mathcal{H}$. It is shown that every gaussian symmetry admits aKlauder-Bargmann integral representation in terms of coherent states.Furthermore, gaussian symmetries, gaussian states and second quantizationcontractions, all of these operators belong to a weakly closed, selfadjointsemigroup $\mathcal{E}_2(\mathcal{H})$ of bounded operators in$\Gamma(\mathcal{H})$. This yields, a new parametrization of gaussian states,which is a very fruitful alternative to the customary parametrization byposition-momentum mean vectors and covariance matrices. This leads to a richharvest of corollaries:  (i) every gaussian state $\rho$ admits a factorization $\rho =Z_{1}^{\dagger}Z_{1}$, where $Z_{1}$ is an element of$\mathcal{E}_2(\mathcal{H})$ and has the form $Z_{1} =\sqrt{c}\Gamma(\sqrt{\Lambda})\exp{\sum_{r=1}^{n} \lambda_ra_r+\sum_{r,s=1}^{n}\alpha_{rs}a_{r}a_{s}}$ on the dense linear manifold generated by allexponential vectors, $\Lambda$ being a positive operator in $\mathcal{H}$,$a_{r}, 1\leq r \leq n$ are the annihilation operators corresponding to the $n$different modes in $\Gamma(\mathcal{H})$, $\lambda_r\in \mathbb{C}$ and$[\alpha_{rs}]$ is a symmetric matrix in $M_n(\mathbb{C})$;  (ii) an explicit particle basis expansion of an arbitrary mean zero puregaussian state vector along with a density matrix formula for a generalgaussian state in terms of its $\mathcal{E}_2(\mathcal{H})$-parameters;  (iii) an easy test for the entanglement of pure gaussian states and a classof examples of pure $n$-mode gaussian states which are completely entangled;  (iv) tomography of an unknown gaussian state in $\Gamma(\mathbb{C}^n)$ by theestimation of its $\mathcal{E}_2(\mathbb{C}^n)$-parameters using $O(n^2)$measurements with a finite number of outcomes.

A Common Parametrization for Finite Mode Gaussian States, their Symmetries and associated Contractions with some Applications